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A127472 Triangle T(n,k) = Sum_{j=k..n, j|n, k|j} phi(j) read by rows, 1<=k<=n.

%I A127472 #7 Aug 12 2015 21:17:06
%S A127472 1,2,1,3,0,2,4,3,0,2,5,0,0,0,4,6,3,4,0,0,2,7,0,0,0,0,0,6,8,7,0,6,0,0,
%T A127472 0,4,9,0,8,0,0,0,0,0,6,10,5,0,0,8,0,0,0,0,4,11,0,0,0,0,0,0,0,0,0,10,
%U A127472 12,9,8,6,0,6,0,0,0,0,0,4,13
%N A127472 Triangle T(n,k) = Sum_{j=k..n, j|n, k|j} phi(j) read by rows, 1<=k<=n.
%C A127472 Defined by the matrix product A054522 * A051731.
%F A127472 T(n,k) = Sum_{j=k..n} A054522(n,j) * A051731(j,k), 1<=k<=n.
%e A127472 First few rows of the triangle are;
%e A127472 .1;
%e A127472 .2, 1;
%e A127472 .3, 0, 2;
%e A127472 .4, 3, 0, 2;
%e A127472 .5, 0, 0, 0, 4;
%e A127472 .6, 3, 4, 0, 0, 2;
%e A127472 .7, 0, 0, 0, 0, 0, 6;
%e A127472 .8, 7, 0, 6, 0, 0, 0, 4;
%e A127472 ....
%p A127472 A127472 := proc(n,k)
%p A127472         a := 0 ;
%p A127472         for j from k to n do
%p A127472                 if (n mod j = 0 ) and (j mod k =0 ) then
%p A127472                         a := a+numtheory[phi](j) ;
%p A127472                 end if;
%p A127472         end do;
%p A127472         a ;
%p A127472 end proc:
%p A127472 seq(seq(A127472(n,k),k=1..n),n=1..14) ; # _R. J. Mathar_, Nov 11 2011
%Y A127472 Cf. A054522, A051731, A062949 (row sums), A000010 (diagonal n=k), A127471 (swapped matrix product).
%K A127472 nonn,tabl
%O A127472 1,2
%A A127472 _Gary W. Adamson_, Jan 15 2007